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John A. Schuster: Draft Articles for Cambridge Descartes Lexicon

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John A. Schuster: Draft Articles for Cambridge

Descartes Lexicon (Ed. Lawrence Nolan)

• Hydrostatics

• Light

• Magnetism

• Physico-Mathematics

• Vortex


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“Hydrostatics”

Hydrostatics was one of a number of areas of “mixed mathematics” —including

geometrical optics, positional astronomy, harmonics and mechanics—developed

by Alexandrian authors in the Hellenistic era. Until the late sixteenth century the

canonical work on hydrostatics was “On Floating Bodies” by Archimedes (c. 287

– 212 BCE). It deals in a rigorous geometrical manner with the conditions under

which fluids are at rest in statical equilibrium, and with the equilibrium conditions

of solid bodies floating in or upon fluids.

At the end of 1618, the twenty-two year old Descartes, working with Isaac

Beeckman, addressed a number of problems in hydrostatics involving the “hydro-

static paradox.” In 1586, Simon Stevin, the leading exponent of the mixed mathe-

matical sciences at the time, brilliantly extended Archimedean hydrostatics. He

demonstrated that a fluid filling two vessels of equal base area and height exerts

the same total pressure on the base, irrespective of the shape of the vessel and

hence, paradoxically, independently of the amount of fluid it contains. Stevin’s

mathematically rigorous proof applied a condition of static equilibrium to various

volumes and weights of portions of the water. (Stevin 1955-66, vol. I, 415-7)

In Descartes’ treatment of the hydrostatic paradox (AT X, 67-74) the key problem

involves vessels B and D, which have equal areas at their bases, equal height and

are of equal weight when empty. (see Figure 1) Descartes proposes to show that,

“the water in vessel B will weigh equally upon its base as the water in D upon its

base”—Stevin’s hydrostatic paradox. (AT X, 68-69 )

Fig. 1. Descartes, “Hydrostatics Manuscript” (AT X 69)

First Descartes explicates the weight of the water on the bottom of a vessel as the

total force of the water on the bottom, arising from the sum of the pressures ex-

erted by the water on each unit area of the bottom. This “weighing down” is ex-

plained as “the force of motion by which a body is impelled in the first instant of

its motion,” which, he insists, is not the same as the force of motion which “bears

the body downward” during the actual course of its fall. (AT X, 68)

In contrast to Stevin’s rigorous proof, Descartes attempts to reduce the phenome-

non to corpuscular mechanics by showing that the force on each “point” of the


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bottoms of the basins B and D is equal, so that the total force is equal over the two

equal areas, which is Stevin’s paradoxical result. He claims that each “point” on

the bottom of each vessel is serviced by a unique line of instantaneously exerted

“tendency to motion” propagated by contact pressure from a point (particle) on the

surface of the water through the intervening particles. But, while the surface of D

is equal to and congruent with its base and posed directly above it, the surface of

B is implied to have an area one-third its base. Hence, exemplary points i, D and l

in the base of D are each pressed by a unique, vertical line of tendency emanating

respectively from corresponding points, m, n and o on the surface. In contrast,

point f on the surface of B is the source of three simultaneous lines of tendency,

two being curved, servicing exemplary points g, B and h on the bottom of B. Des-

cartes claims that all six exemplary points on the bottoms are pressed by an equal

force, because they are each pressed by “imaginable lines of water of the same

length” (AT X, 70); that is, lines having the same vertical component of descent.

Descartes smuggles the tendentious three-fold mapping from f into the discussion

as an “example” but then argues that given the mapping, f can provide a three-fold

force to g, B and h. (AT X, 70-1)

The “hydrostatic manuscript” shows the young Descartes articulating the program

that he and Beeckman at that time termed “physico-mathematics,” in which reli-

able geometrical results in the mixed mathematical sciences were to be explained

by an embryonic corpuscular-mechanical matter theory and causal discourse con-

cerning forces and tendencies to motion. Stevin’s treatment of the hydrostatic

paradox is within the domain of mixed mathematics, rather than natural philoso-

phy. It does not explain the phenomenon by identifying its causes. Descartes’ ac-

count falls within the domain of natural philosophy, attempting to identify the ma-

terial bodies and causes in play. For Descartes fluids are made up of corpuscles

whose tendencies to movement are understood in terms of a theory of forces and

tendencies. This can explain the pressure a fluid exerts on the floor of its contain-

ing vessel. These moves imply a radically non-Aristotelian vision of how the

mixed mathematical sciences, such as hydrostatics, should relate to natural phi-

losophising. (Gaukroger and Schuster 2002, 549-550)

Although Descartes never again directly considered hydrostatical problems, this

early fragment is of tremendous importance for understanding his mature natural

philosophy. Throughout his later career Descartes continued to use descendants of

the concept of instantaneous tendency to motion analysable into its directional

components (later termed “determinations”). (see force) These ideas are central to

his “dynamics,” the concepts that govern the behaviour of micro-corpuscles in the

Treatise on Light and the Principles of Philosophy. (see light and vortex)

See: BEECKMAN, ISAAC; EARLY WRITINGS; FORCE; LIGHT;

MECHANICS; PHYSICO–MATHEMATICS; PRINCIPLES OF PHILOSOPHY;

TREATISE ON LIGHT; VORTEX


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For Further Reading

Primary Source

Stevin, Simon. 1955-66. The Principal Works of Simon Stevin, ed. Ernst Cronie et al., 5 vols.

Amsterdam: Swets & Zeitlinger.

Secondary Sources

Gaukroger, Stephen and J. A. Schuster. 2002. “The Hydrostatic Paradox and the Origins of Car-

tesian Dynamics,” Studies in the History and Philosophy of Science 33: 535-572.

Gaukroger, Stephen. 2000. “The Foundational Role of Hydrostatics and Statics in Descartes’

Natural Philosophy,” in S. Gaukroger, J. Schuster, and J. Sutton, eds., Descartes’ Natural

Philosophy. London: Routledge, 60-80.


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“Light”

Descartes’ corpuscular-mechanical natural philosophy is intended to replace the

Aristotelianism of the late medieval universities and the resurgent neo-Platonic

natural philosophies, in which light is conceived as the intermediary between base

matter and higher spiritual and immaterial entities. In the most simple version of

his theory, Descartes explains light mechanically as a tendency to motion, an im-

pulse, propagated instantaneously through continuous optical media. This has the

very important implication that in Descartes’ theory, the propagation of light is in-

stantaneous, but the magnitude of the force conveyed by the tendency to motion

constituting light can vary—there can be stronger and weaker light rays, all propa-

gated instantly. (Schuster 2000, 261)

Descartes’ theory of light cannot be understood in detail without his theory of cor-

puscular dynamics. (see force) Descartes holds that bodies in motion, or tending

to motion, are characterised from moment to moment by the possession of two

sorts of dynamical quantity: (1) the absolute quantity of the ‘force of motion’ and

(2) the directional modes of that quantity of force, which Descartes calls “deter-

minations.” As corpuscles undergo instantaneous collisions, their quantities of

force of motion and determinations alter according to the laws of nature. Des-

cartes focuses on instantaneous tendencies to motion, rather than finite translations

in space and time. His exemplar for applying these concepts is the dynamics of a

stone rotated in a sling. (see figure 1) (AT XI 45-6, 85; G 30, 54-5)

Figure 1: Descartes’ Dynamics of the Sling in Treatise on Light

Descartes considers the stone at the instant that it passes point A. The instantane-

ously exerted force of motion of the stone is directed along the tangent AG. If the

stone were released and nothing affected its trajectory, it would move along ACG

at a uniform speed reflective of its conserved quantity of force of motion. How-


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ever, the sling continuously constrains what can be termed the ‘principal’ determi-

nation of the stone and, acting over time, deflects its motion along the circle AF.

The other component of determination acts along AE, completely opposed by the

sling, so that only a tendency to centrifugal motion occurs, rather than centrifugal

translation. It is this conception of centrifugal tendency that Descartes uses when

he articulates his theory of light inside his cosmological theory of vortices.

Each vortex consists of a central star, made up of the highly agitated and ex-

tremely small particles of first element, surrounded by a fluid which rotates

around the star. The fluid consists of spherical particles of second element, which

are significantly larger than first element particles, and which are tightly packed

together, maintaining points of contact with each other. The remaining interstitial

spaces are filled with first element particles. As the fluid rotates, the particles of

second element generate centrifugal tendency to recede from the center, but since

they are tightly packed together and every vortex is bounded by other vortices, this

tendency cannot manifest itself as actual centrifugal translation of the second ele-

ment particles. Descartes identifies light with the resulting lines, or rays, of cen-

trifugal tendency to motion transmitted instantaneously outward through the vor-

tex. (AT XI 87-90; G 55-8; AT VIIIA 108-116; MM 111-118)

In his Optics of 1637 Descartes presented the long sought after law of refraction

of light, and attempted to demonstrate it (see figure 2) by use of a model of a ten-

nis ball struck by a racket along AB towards refracting surface CBE. (AT VI 97-8;

CSM I 158-9) Using Descartes’ theory of light, and his corpuscular dynamics, one

can analyse both his published “proof” of the law of refraction, and its underlying

rationale in terms of his real theory of light as instantaneous tendency to motion

transmitted through spherical particles of second element. The tennis ball’s weight

and volume are ignored. It moves without air resistance in empty geometrical

space on either side of the cloth, which is taken to be perfectly flat and vanishingly

thin. In breaking through the cloth, the ball loses, independently of its angle of in-

cidence, a certain fraction (one half) of its total quantity of force of motion. Des-

cartes applies two conditions to the motion of the ball: [a] the new quantity of

force of motion is conserved during motion below the cloth; and [b] the parallel

component of the force of motion, the parallel determination, is unaffected by the

encounter with the cloth. Drawing a circle of radius AB around B, he assumes the

ball took time t to traverse AB prior to impact. After impact, losing half of its

force of motion, hence half its speed, it must take 2t to traverse a distance equal to

AB, arriving somewhere on the circle after 2t. This represents condition [a]. Des-

cartes writes that prior to impact the parallel determination “caused” the body to

move towards the right between lines AC and HBG. For condition [b] he consid-

ers that after impact, the ball takes 2t to move to the circle's circumference, so its

unchanged parallel determination has twice as much time in which to act to

“cause” the ball to move toward the right. He sets FEI parallel to HBG so as to

represent that doubled parallel travel. At time 2t after impact the ball will be at I,


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the intersection of FEI and the circle point below the cloth. It follows that (sin i/sin

r) = (AH/KI) or 0.5 for all angles of incidence.

Figure 2: Descartes’ Figure for Refraction of Light (Tennis Ball) in Optics

Descartes’ published proof is superficially kinematic. But if we consider his cor-

puscular dynamics and the fact that his tennis ball is virtually a mathematical point

in motion, we can translate Descartes’ proof into the terms of his actual theory of

light as instantaneously propagated tendency to motion. (see figure 3) Consider a

light ray incident upon refracting surface CBE. Let length AB represent the mag-

nitude of the force of the light impulse. The orientation and length of AB repre-

sent the principal determination of the ray. The force of the ray is diminished by

half in crossing the surface. So, to represent condition [a] we draw a semi-circle

below the surface about B with a radius equal to one half of AB. As for condition

[b], the unchanged parallel determination, we simply set out line FEI parallel to

HBG and AC so that AH=HF. The resulting intersection at I gives the new orien-

tation and magnitude of the force of the ray of light, BI and the law follows, as a

law of cosecants. The case of the light ray requires manipulation of unequal semi-

circles, representing the ratio of the force of light in the two media. In the tennis

ball case Descartes moves from the ratio of forces to the ratio of speeds and hence

the differential times to cross equal circles. But, at the instant of impact, the same

force and determination relations are attributed to the tennis ball and the light ray.


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Figure 3: Refraction of Light Using Descartes’ Dynamics and Real Theory of Light

The derivation of the long sought law of refraction using the principles of his dy-

namics of corpuscles marked the high point of Descartes’ optical researches, along

with his application of the law to the explanation of the telescope and to an ingen-

ious solution of the equally long standing problem of explaining the rainbow.

(also see optics) However, the full meaning of Descartes’ optical triumph in rela-

tion to the overall development of his corpuscular-mechanical natural philosophy

can only be grasped by looking at how his optical work unfolded over time, start-

ing with his discovery of the law of refraction in Paris in 1626/7 while collaborat-

ing with the mathematician Claude Mydorge. This was accomplished independ-

ently of, but in the same manner as, Thomas Harriot, who first discovered the law

around 1598. (see figure 4)


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Figure. 4 Thomas Harriot’s Key Diagram

Harriot used the traditional image locating rule to map the image locations of

point sources taken on the lower circumference of a half-submerged disk refrac-

tometer. (Lohne, 1963; Buchdahl 1972) This yielded a smaller semi-circle as the

locus of image points and hence a cosecant law of refraction of light. In a letter

describing an identical cosecant form of the law, Mydorge presents a virtually

identical diagram (see figure 5), but moves the inner semi-circle above the inter-

face as a locus of point sources for the incident light. (Mersenne, 1932-88, I 404-

15; Schuster 2000, 272-3)

Figure 5 Mydorge’s (and Descartes’) Cosecant form of the law of refraction


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Figure 5 closely resembles Figure 3, the derivation of the law of refraction using

Descartes’ conditions from the Dioptrique and his theory of light as instantane-

ously propagated tendency to motion. It is the key to unpacking the co-evolution

of Descartes’ theory of light and his dynamics of corpuscles.

After his discovery of the law of refraction by these purely geometrical optical

means, issuing in the cosecant form of the law, Descartes sought to explain the

law by use of a dynamics of corpuscles. Working in the style of his “physico-

mathematics,” he transcribes into dynamical terms some of the geometrical pa-

rameters embodied in the cosecant representation. The resulting dynamical princi-

ples concerning the mechanical nature of light are: [1] the absolute quantity of the

force of the ray is increased or decreased in a fixed proportion, while [2] the paral-

lel component of the force of a light ray is unaffected by refraction. These are ef-

fectively the conditions [a] and [b] which control the derivation of the law of re-

fraction in the Dioptrique, and this is how he arrived at them, as is confirmed by

the fact that by late 1628, Descartes used these concepts to explain the law of re-

fraction to his friend Isaac Beeckman. (AT X, 336; Schuster 2000, 290-95)

So, principles [1] and [2] that control the proof of the law of refraction in the real

theory of light and in the tennis ball model of light, were abstracted from the

original geometrical representation of the newly discovered cosecant form of the

law. Furthermore, these two mechanistic conditions for a theory of refraction sug-

gested the two central tenets of Descartes’ mature dynamics as he composed the

Treatise on Light (1629-33). The first rule of nature in the Treatise on Light as-

serts the conservation of the quantity of the instantaneously exerted force of mo-

tion of a body in the absence of external causes. This rule subsumes and general-

ises [1]. The third rule of nature defines what we above called the principle

determination of the instantaneously exerted force of motion of a body, along the

tangent to the path of motion at the instant under consideration. This rule thus sub-

sumes and generalises [2].

In sum, for Descartes the basic laws of light—itself an instantaneously transmitted

mechanical impulse—immediately revealed the principles of the instant-to-instant

dynamics of corpuscles. Not only is the theory of light central to the elaboration of

Descartes’ entire mechanistic system, but the principles of the dynamics of cor-

puscles governing that system arise from his research in geometrical and then

mechanistic optics.

See BEECKMAN, ISAAC; COLOR; DIOPTRICS; FORCE; LAWS OF

NATURE; MECHANICS; METEOROLOGY; MYDORGE, CLAUDE; OPTICS;

PHYSICO–MATHEMATICS; PHYSICS; PRINCIPLES OF PHILOSOPHY;

RAINBOW; TREATISE ON LIGHT; VORTEX


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Further Reading

Primary Source

Mersenne, Marin. 1932-88. Correspondence du P. Marin Mersenne. Cornelius de Waard, R. Pin-

tard, B. Rochot, and A. Baelieu, eds., 17 vols. Paris: Centre National de la Recherche Scienti-

fique.

Secondary Sources

Buchdahl, Gerd. 1972. “Methodological Aspects of Kepler's Theory of Refraction”, Studies in

the History and Philosophy of Science 3: 265-98.

Lohne, Johannes. 1963. “Zur Geschichte des Brechungsgesetzes,” Sudhoffs Archiv 47: 152-72.

Sabra, A. I. 1967. Theories of Light from Descartes to Newton. London, Oldbourne.

Schuster, John. A. 2000. “Descartes opticien: The Construction of the Law of Refraction and the

Manufacture of its Physical Rationales,” in Stephen Gaukroger, John Schuster, and John Sut-

ton, eds., Descartes’ Natural Philosophy. London: Routledge, 258-312.


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“Magnetism”

Magnetism, long considered the exemplar of an occult, spiritual power, posed a

challenge to mechanical philosophers like Descartes. William Gilbert’s De Mag-

nete (1600) offered an impressive natural philosophy, grounded in experiments,

that could lead to interpreting magnetism as an immaterial power that possesses in

its higher manifestations the capabilities of soul or mind. In his Principles of Phi-

losophy Descartes accepts Gilbert’s experiments, but he explains magnetism

mechanistically, based on the movements of two species, right– and left–handed,

of “channelled” or cylindrical screw-shaped particles of his first element. Des-

cartes claims that magnetic bodies—naturally occurring lodestone, or magnetised

iron or steel—have two sets of pores running axially between their magnetic

poles: one set accepts only right–handed channelled particles; the other set of

pores accepts only the left–handed particles. Descartes thus explained Gilbert’s

experiments, including his use of a sphere of loadstone, to demonstrate the proper-

ties of magnetised compass needles.

However, Descartes did more than appropriate and reinterpret Gilbert’s “labora-

tory” work. Gilbert called his sphere of lodestone a terrella, a “little earth,” argu-

ing that because compass needles behave identically on the terrella as on the earth

itself, the earth is, essentially, a magnet. Hence, according to his natural philoso-

phy, the earth possesses a magnetic “soul”, capable of causing it to spin. Magnetic

“souls” similarly cause the motions of other heavenly bodies. In his Principles,

Descartes, aiming to displace Gilbert’s natural philosophy, focuses on the “cos-

mic” genesis and function of his channelled magnetic particles. Descartes argues

that the spaces between the spherical corpuscles of the second element that make

up his vortices, are roughly triangular, so that particles of the first element, con-

stantly being forced through the interstices of second element spheres, become

“channelled” or “grooved” with triangular cross–sections. Such first element cor-

puscles tend to be flung by centrifugal tendency out of the equatorial regions of

vortices and into neighboring vortices along the north and south directions of their

axes of rotation, thus receiving opposite axial twists. The resulting left– and right–

handed screw shaped first element particles penetrate into the polar regions of cen-

tral stars and then bubble up toward their surfaces to form, Descartes claims, sun

spots. Stars are thus magnetic, as Gilbert maintained, but in a mechanistic sense.

Moreover, for Descartes, planets are also magnetic, as Gilbert claimed, but again

the explanation is mechanical. Descartes describes how a star may become totally

encrusted by sun spots. This extinguishes the star, its vortex collapses and it is

drawn into a neighboring vortex to orbit its central star as a planet. But, such plan-

ets, including our earth, bear the magnetic imprint of their stellar origins, by pos-

sessing axial channels between their magnetic poles accommodated to the right- or

left-handed screw particles. Descartes’ explanation ranges from the cosmic pro-

duction of magnetic particles, through the nature of stars and sunspots, to the birth


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and history of planets. He accepts the cosmic importance of magnetism, but ren-

ders the explanation mechanical, thus binding his natural philosophy into a cos-

mogonical and cosmological whole.

See COSMOLOGY; ELEMENT; EXPERIMENT; PRINCIPLES OF

PHILOSOPHY; SUBTLE MATTER; VORTEX

For Further Reading:

Shea, William. 1991. The Magic of Numbers and Motion: The Scientific Career of René Des-

cartes. Canton: Science History Publications.

Gaukroger, Stephen. 2002. Descartes’ System of Natural Philosophy. Cambridge: CUP.


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“Physico-Mathematics”

In November 1618, Descartes, then twenty-two, met and worked for two months

with Isaac Beeckman, a Dutch scholar eight years his senior. Beeckman was one

of the first supporters of a corpuscular–mechanical approach to natural philoso-

phy. However, it was not simply corpuscular mechanism that Beeckman advo-

cated to Descartes. He also interested Descartes in what they called physico-

mathematics. In late 1618, Beeckman wrote that “There are very few physico-

mathematicians,” adding, “(Descartes) says he has never met anyone other than

me who pursues enquiry the way I do, combining Physics and Mathematics in an

exact way; and I have never spoken with anyone other than him who does the

same.” (Beeckman 1939-53, I, p.244) They were partly right. While there were

not many physico-mathematicians, there were of course others, such as Kepler,

Galileo and certain leading Jesuit mathematicians, who were trying to merge

mathematics and natural philosophy. (Dear 1995, 168-79)

Physico-mathematics, in Descartes’ view, deals with the way the traditional mixed

mathematical disciplines, such as hydrostatics, statics, geometrical optics, geo-

metrical astronomy, and harmonics, were conceived to relate to the discipline of

natural philosophy. In Aristotelianism, the mixed mathematical sciences were in-

terpreted as intermediate between natural philosophy and mathematics and sub-

ordinate to them. Natural philosophical explanations were couched in terms of

matter and cause, something mathematics could not offer, according to most

Aristotelians. In the mixed mathematical sciences, mathematics was used not in an

explanatory way, but instrumentally for problem solving and practical aims. For

example, in geometrical optics, one represented light as light rays. This might be

useful but does not facilitate answering the underlying natural philosophical ques-

tions: “the physical nature of light” and “the causes of optical phenomena.” In

contrast, physico-mathematics involved a commitment to revising radically the

Aristotelian view of the mixed mathematical sciences, which were to become

more intimately related to natural philosophical issues of matter and cause. Para-

doxically, the issue was not mathematization. The mixed mathematical sciences,

which were already mathematical, were to become more “physicalized,” more

closely integrated into whichever brand of natural philosophy an aspiring physico-

mathematician endorsed.

Three of Descartes’ exercises in physico-mathematics survive. The most important

is his attempt, at Beeckman’s urging, to supply a corpuscular-mechanical explana-

tion for the hydrostatic paradox, which had been rigorously derived in mixed

mathematical fashion by Simon Stevin. (AT X 67-74, 228; Gaukroger and Schus-

ter, 2002) Descartes’ physico-mathematical work on hydrostatics involves a radi-

cally non–Aristotelian vision of the relation of the mixed mathematical sciences to

his emergent form of corpuscular mechanical natural philosophy. Descartes aims

to shift hydrostatics from mixed mathematics into the realm of natural philosophy.


15

He believed that from crisp, simple geometrical representations of sound mixed

mathematical results one can read out or “see” the underlying corpuscular me-

chanical causes.

Descartes and Beeckman’s studies of accelerated free fall also belong to their

physico–mathematical project. They did not achieve any agreed results, because

they could neither settle on what the correct, geometrically expressed law of fal-

ling bodies is, nor discern firm clues about its underlying causes. (AT X 58-61,

74-78, 219-222) The failed physico-mathematicization of falling bodies reverber-

ates later in Descartes’ distrust of the scientific relevance of Galileo’s announce-

ment of a mathematical law of accelerated free fall.

Descartes’ third physico–mathematical exercise showed more promise, stalling in

the short run, although it yielded rich results later. In 1620 he attempted in a phys-

ico–mathematical manner to find the law of refraction of light by considering the

geometrical representation of its likely causes. He based the endeavour on pas-

sages and diagrams in which Kepler suggests that light moves with more force in

denser optical media and “hence” is bent toward the normal in moving from a less

to a more dense medium. (AT X 242-3) On this occasion, Descartes found neither

a law of refraction nor its natural philosophical causes. However, seven years

later, while working with the mathematician Claude Mydorge, he found, by tradi-

tional mixed mathematical means, a simple (cosecant) version of the law of refrac-

tion. Descartes immediately set to work attempting, in a physico-mathematical

manner, to read out of his key geometrical diagram the principles of a mechanical

theory of light that would then subsume the new geometrical law that had

prompted them. These developments in turn had large consequences for the sys-

tem of corpuscular-mechanical natural philosophy he first developed in his Trea-

tise on Light (1629-33): Descartes’ ideas about mechanistic optics, themselves

physico-mathematical in tenor, suggested key concepts of his dynamics of corpus-

cles, which in turn helped shape his theory of vortices.

See: BEECKMAN, ISAAC; EARLY WRITINGS;

HYDROSTATICS; LIGHT; MECHANICS; VORTEX

For Further Reading

Primary Source

Beeckman, Isaac. 1939-53. Journal tenu par Isaac Beeckman de 1604 à 1634, ed. Corne-

lius de Waard, 4 vols. The Hague: Nijhof.

Secondary Sources

Dear, Peter. 1995. Discipline and Experience. Chicago: University of Chicago Press.

Gaukroger, Stephen and J. A. Schuster. 2002. “The Hydrostatic Paradox and the Origins of

Cartesian Dynamics,” Studies in the History and Philosophy of Science 33: 535-572.


16

Jullien, Vincent and A. Charrak. 2002. Ce que dit Descartes touchant la chute des graves:

de 1618 à 1646, étude d’un indicateur de la philosophie naturelle cartésienne. Ville-

neuve d’Ascq (Nord): Presses Universitaires du Septentrion.

Schuster, John A. 2000. “Descartes opticien: The Construction of the Law of Refraction

and the Manufacture of its Physical Rationales,” in S. Gaukroger, J. Schuster, and J.

Sutton, eds., Descartes’ Natural Philosophy. London: Routledge, 258-312.


17

“Vortex”

The theory of vortical celestial mechanics, as presented in the Principles of Phi-

losophy and Treatise on Light, is the “engine room” of Descartes’ system of natu-

ral philosophy. Descartes starts his vortex theory with an “indefinitely” large

chunk of divinely created matter or extension in which there are no void spaces

whatsoever. When God injects motion into this extension, it is shattered into mi-

cro-particles and myriads of “circular” displacements ensue, forming large num-

bers of gigantic whirlpools or vortices. This process eventually produces three

species of corpuscle, or elements, along with the birth of stars and planets. The

third element forms all solid and liquid bodies on all planets throughout the cos-

mos, including the earth. Interspersed in the pores of such planetary bodies are the

spherical particles of the second element. The second element also makes up the

bulk of every vortex, while the spaces between these spherical particles are filled

by the first element, which also constitutes the stars, including our sun.

The key to Descartes’ celestial mechanics is his concept of the “massiveness” or

“solidity” of a planet, meaning its aggregate volume to surface ratio, which is in-

dicative of its ability to retain acquired motion or to resist the impact of other bod-

ies. The particles of the second element making up a vortex also vary in volume to

surface ratio with distance from the central star, as gathered from Descartes’ stipu-

lations concerning the variation of the size (and speed) of the second element par-

ticles with distance from the central star, illustrated in Figure 1. Note also the im-

portant inflection point in the size and speed curves at radius K. (Schuster 2005,

49) A planet is locked into an orbit at a radial distance at which its centrifugal ten-

dency, related to its aggregate solidity, is balanced by the counter force arising

from the centrifugal tendency of the second element particles composing the vor-

tex in the vicinity of the planet—that tendency similarly depending on the volume

to surface ratio of the those particular particles.

The most massive planet in a star system will be closest to, but not beyond the K

layer—as Saturn is in our planetary system. Comets are planets of such high solid-

ity that they overcome the resistance of the second element particles at all dis-

tances up to and including K. Such an object will pass beyond the K level, where

it will meet second element particles with decreasing volume to surface ratios,

hence less resistance, and be extruded out of the vortex into a neighbouring one.

Entering the neighbouring vortex, the comet falls, and spirals, downward toward

its central star, all the time meeting increasing resistance from the second element

particles above that vortex’s K distance. As it picks up increments of orbital

speed, the comet starts to generate increasing centrifugal tendency, begins to rise

and spiral upward, and eventually is flung back out of the second vortex.


18

Fig.1 Size, Speed & Force of Motion Distribution Of Particles Of 2nd

Element, In A Stellar

Vortex

Also essential to Descartes’ theory is a principle of vortex stability, which he in-

troduces using his ideas about the dynamics governing the motions of corpuscles

(light). In the early stages of vortex formation, before stars and elements have

evolved, the then existing vortical particles become arranged so that their centrifu-

gal tendency increases continuously with distance from the center. (AT XI 50-1; G

33) As each vortex settles out of the original chaos, the larger corpuscles are

harder to move, resulting in the smaller ones acquiring higher speeds. Hence, in

these early stages, the size of particles decreases and their speed increases from

the center out. But the speed of the particles increases proportionately faster, so

that force of motion (size times speed) increases continuously. Figure 2 shows the

distribution of size and speed of the particles in any vortex before a central star

and the three elements have formed. (Schuster 2005, 46 )


19

Figure 2. Size, Speed and Force of Motion Distribution of Particles of 2nd

Element, Prior to

Existence of Central Star

Stars do not exist in the early stages of vortex formation as described by Des-

cartes. They form in the center of each vortex as part of the process leading to the

emergence of the three final Cartesian elements. Every star alters the original size

and speed distribution of the particles of the vortex, in a way that now allows

planets to maintain stable orbits. Descartes explains that a star is made of up the

most agitated particles of first element. Their agitation, and the rotation of the star,

communicate extra motion to particles of the vortex near the star’s surface. This

increment of agitation decreases with distance from the star and vanishes at that

key radial distance, called K. (Figure 3) (AT XI, 54-6; G 35-7; Schuster 2005, 48)

Figure 3 Agitation Due To Existence Of Central Star


20

This stellar effect alters the original size and speed distribution of the spheres of

second element in the vortex, below the K layer. There now are greater corpuscu-

lar speeds close to the star than in the pre-star situation. But the all important vor-

tical stability principle still holds, so the overall size/speed distribution must

change, below the K layer. Descartes ends with the situation in figure 1, with the

crucial inflection point at K. Beyond K we have the old (pre-star formation) stable

pattern of size/speed distribution; below K we have a new, (post–star formation)

stable pattern of size/speed distribution. This new distribution turns a vortex into a

machine which, as described, locks planets into appropriate orbits below K and ex-

trudes comets into neighboring vortices. In this way Descartes follows Johannes

Kepler’s lead in attempting to theorize about the physical role played in celestial

mechanics by the sun, or any central star in a planetary system. Copernicus him-

self had never raised the issue of the sun’s causal role in planetary motion.

In its intimate technical design Descartes’ vortex mechanics is a science of equi-

librium, resembling his work on hydrostatics in his early program in physico-

mathematics. (Gaukroger and Schuster 2002; Gaukroger 2000; Schuster 2005)

The forces at work upon a planet can only be fully specified when orbital equilib-

rium has been attained, although, of course no actual measurements are involved.

The radial movement of a planet or comet (its rise or fall in a vortex) results from

the breakdown of equilibrium and cannot be defined mathematically. Despite this

limitation, Descartes intended that his theory of vortices qualitatively unify the

treatment of celestial motions and the phenomena of local fall and of planetary

satellites. A comet extruded from one vortex enters a neighboring one and falls

toward its K layer before picking up centrifugal force and rising again out of the

vortex in question. Similarly, Descartes makes it clear that a planet ‘too high up’

in the vortex for its particular solidity is extruded sun-ward, falling (and spiralling)

down in the vortex to find its proper orbital distance. (AT XI 65-66; G 42; AT

VIIIA 193; MM 169) In exactly the same fashion, Descartes’ theory of local fall

(AT XI 73-4; G 47), and theory of the orbital motion of the moon, when taken in

their simplest acceptations, both also make use of the notion of falling in a vortex

until a proper orbital level is found (assuming no other circumstances prevent

completion of the process, as they do in local fall of heavy terrestrial bodies near

the surface of the earth). However, Descartes’ treatment of locally falling bodies

and the motion of satellites both run into considerable difficulties when he at-

tempts to explicate them in detail. (gravity)

Taken in both its technical details and its qualitative sweep, Descartes' vortex the-

ory was a considerable achievement. The theory signaled Descartes’ commitment

to the truth of Copernicanism writ large, as an account of innumerable star and

planet systems—interlocking sub-systems in the great machine of nature. He pre-

figured Newton in trying to bring planets, comets, satellites and locally falling

bodies within one explanatory web, and his vortex theory persisted into the eight-

eenth century to compete with Newton’s physics. (Aiton 1972)


21

See COSMOLOGY; EARTH, MOTION OF; FORCE; GRAVITY; PHYSICO–

MATHEMATICS; LIGHT; HYDROSTATICS; MECHANICS; PHYSICS;

PRINCIPLES OF PHILOSOPHY; TREATISE ON LIGHT

For Further Reading

Aiton, Eric J. 1972. The Vortex theory of Planetary Motion. London: Macdonald.

Gaukroger, Stephen. 2002. Descartes’ System of Natural Philosophy. Cambridge: CUP.

Gaukroger, Stephen. 2000. “The Foundational Role of Hydrostatics and Statics in Descartes’

Natural Philosophy,” in Stephen Gaukroger, John Schuster, and John Sutton, eds., Descartes’

Natural Philosophy. London: Routledge, 60-80.

Gaukroger, Stephen and John A. Schuster. 2002. “The Hydrostatic Paradox and the Origins of

Cartesian Dynamics,” Studies in the History and Philosophy of Science 33: 535-572.

Schuster, John A. 2005. “Waterworld: Descartes’ Vortical Celestial Mechanics—A Gambit in

the Natural Philosophical Contest of the Early Seventeenth Century’, in Peter Anstey and J.A.

Schuster, eds., The Science of Nature in the Seventeenth Century: Patterns of Change in

Early Modern Natural Philosophy. Kluwer/Springer: Dordrecht, pp. 35-79.

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